Problem: Let $C$ be the line defined by $3x - 5y = 2$. We have a change of variables: $\begin{aligned} x &= X_1(u, v) = u^2 - 4v^2 \\ \\ y &= X_2(u, v) = -3u^2 + 2v^2 \end{aligned}$ What is $C$ under the change of variables? Choose 1 answer: Choose 1 answer: (Choice A) A $16u^2 - 19v^2 = 1$ (Choice B) B $4u^2 - 15v^2 = 1$ (Choice C) C $8u^2 - 5v^2 = 1$ (Choice D) D $9u^2 - 11v^2 = 1$
Explanation: When applying a change of variables, we substitute the new definition for $x$ and $y$ into the original equation. The original equation: $3x - 5y = 2$ Let's substitute $X_1(u, v)$ for $x$ and $X_2(u, v)$ for $y$. $\begin{aligned} 3 \left( u^2 - 4v^2 \right) - 5 \left( -3u^2 + 2v^2 \right) &= 2 \\ \\ 3u^2 - 12v^2 + 15u^2 - 10v^2 &= 2 \\ \\ 18u^2 - 22v^2 &= 2 \\ \\ 9u^2 - 11v^2 &= 1 \end{aligned}$ Therefore, under the change of variables, $C$ becomes: $9u^2 - 11v^2 = 1$